Question 105128
First off, I'm changing variables because 2 and Z look too much alike and it gets confusing trying to follow. 
{{{u + 1/u = 1}}}
Let's look at {{{(u+1/u)^2}}}
{{{(u+1/u)^2=(u+1/u)(u+1/u)}}}
Using the FOIL method
{{{(u+1/u)^2=u^2+u(1/u)+(1/u)u+1/u^2}}}
{{{(u+1/u)^2=u^2+2+1/u^2}}}
But from your original equation, you know that
{{{(u+1/u)=1}}}
{{{highlight((u+1/u))^2=u^2+2+1/u^2}}}
{{{highlight(1)^2=u^2+2+1/u^2}}}
1.{{{u^2+1/u^2=-1}}}
Next, look at {{{(u^2+1/u^2)^2}}}
{{{(u^2+1/u^2)^2=(u^2+1/u^2)(u^2+1/u^2)}}}
Using the FOIL method
{{{(u^2+1/u^2)^2=u^4+u^2(1/u^2)+(1/u^2)u^2+1/u^4}}}
{{{(u^2+1/u^2)^2=u^4+2+1/u^4}}}
From 1 you know,
1.{{{u^2+1/u^2=-1}}}
{{{highlight((u^2+1/u^2))^2=u^4+2+1/u^4}}}
{{{highlight((-1))^2=u^4+2+1/u^4}}}
{{{1=u^4+2+1/u^4}}}
2.{{{u^4+1/u^4=-1}}}
You see a pattern here. 
Next, look at {{{(u^4+1/u^4)^2}}}
{{{(u^4+1/u^4)^2=(u^4+1/u^4)(u^4+1/u^4)}}}
{{{(u^4+1/u^4)^2=u^8+u^4(1/u^4)+(1/u^4)u^4+1/u^8}}}
{{{(u^4+1/u^4)^2=u^8+2+1/u^8}}}
From 2 you know,
2.{{{u^4+1/u^4=-1}}}
{{{highlight((u^4+1/u^4))^2=u^8+2+1/u^8}}}
{{{highlight(-1)^2=u^8+2+1/u^8}}}
{{{1=u^8+2+1/u^8}}}
3.{{{u^8+1/u^8=-1}}}
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Let's fast forward.
I'll bet that you can get to this answer by continuing to follow the steps that we followed up until now to get the answer. 
{{{u^64+1/u^64=-1}}}